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Let G be a finite group and H ≤ G. The authors study the structure of finite groups G having a subgroup H which is maximal in 〈H, Hg〉 for some g ∈ G. Some results on the structure of 〈H, Hg〉 and G are set up. Especially, a generalization of Baer′s theorem is established.

We introduce a new subgroup embedding property of finite groups called CSQ-normality of subgroups. Using this subgroup property, we determine the structure of finite groups with some CSQ-normal subgroups of Sylow subgroups. As an application of our results, some recent results are generalized.

In this paper, we investigate the influence of weakly s-semipermutable subgroups on the p F -hypercenter of a finite group and obtained some new criteria for a normal subgroup to be contained in the p F -hypercenter of a finite group. Many known results are generalized and uniformed. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is called an ICΦs -subgroup of G if (H ∩ [ H , G ]) H G / H G ≤ Φ (H / H G) H s G / H G , where H G = ∩ g ∈ G H g and Φ (H / H G) is the Frattini subgroup of H / H G and HsG is the largest s-permutable subgroup of G contained in H. In this paper, we investigate the structure of finite group G under the assumption that some subgroups of G are ICΦs -subgroups of G, some results are obtained. [ABSTRACT FROM AUTHOR]

A subgroup S of G is called a p-sylowizer of a p-subgroup R in G if S is maximal in G with respect to having R as its Sylow p-subgroup. In this paper, we give some new characterizations of p-solvable and p-supersolvable groups by the p-cover-avoidance properties of the p-sylowizers of some p-subgroups and the p-sylowizers in some classes of groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group, p a prime, P a Sylow p-subgroup of G and d a power of p such that 1 < d < | P |. Let G p * denote the unique smallest normal subgroup of G for which the corresponding factor group is abelian of exponent dividing p – 1. Let F 1 , F 2 , F 3 be classes of all p-groups, p-nilpotent groups and p-supersolvable groups, respectively, G F be the F -residual of G. Let E ∈ { G F 1 , (G p *) F 1 , G F 2 , G F 3 }. We prove that if H ∩ E is weakly s-semipermutable in G for all normal subgroups H of P with | H | = d , then either G ∈ F 3 and G ∈ F 2 if p is the least prime factor of | G | , or else | P ∩ E | > d ; and for all subgroups H of P with | H | = d and 4 (if p = 2), then G ∈ F 3 and G ∈ F 2 if p is the least prime factor of | G |. Many known theorems become Corollaries of our results. Communicated by Peter Sin [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is said to satisfy the $${\Pi}$$ - property in G if every prime dividing $${|G : {\rm N}_{G}(HK \cap L)|}$$ also divides the order of $${(HK \cap L)/K}$$ , for every G-chief factor L/ K of G. The $${\Pi}$$ -property is a subgroup embedding property of arithmetical character introduced in Li (J. Algebra 334:321-337, ) that gathers common properties of many other well-known subgroup embeddings (most of them of normal type) and allows us to glimpse common behaviors to all of them. The aim of this note is to prove that a finite group G is soluble if and only if in G all maximal subgroups satisfy the $${\Pi}$$ -property in G. This is the answer to a question posed by Li (J. Algebra 334:321-337, , Question 5.2). [ABSTRACT FROM AUTHOR]

Let G be a finite p-group. If |CG(x): ⟨ x ⟩ | ≤pkfor every noncentral element x of G, then G is called a k − 𝒩𝒞 −p-group. In this article, we give a classification of k − 𝒩𝒞 −p-groups, where k = 1, 2 and p > 2. [ABSTRACT FROM PUBLISHER]

Let G be a finite group and p a prime. In this article, we give a criterion of p-nilpotency of G by means of the normalizer and a generalization of normality of subgroups. [ABSTRACT FROM PUBLISHER]

In this article, we classify p-groups whose non-normal subgroups have index p in their normalizers. This answers a problem posed by Berkovich and Janko. [ABSTRACT FROM AUTHOR]